Our continuum model goes beyond the BM model in several aspects. First, the p-h symmetry of the narrow bands is only weakly broken within the BM model, while it is much more strongly broken in our continuum theory constructed for the tight binding model of Ref. [1]}. While the p-h symmetry of the energy spectrum is broken in both models [2]}, [1]}, we focused on the p-h asymmetry of the narrow band Hilbert space, because it is more important in determining the correlated ground states near the magic angle. As shown in Fig. REF , the p-h asymmetry is dominated by the contribution from the inter-layer contact term \(w_3\)  that has been overlooked in previous works. Another source of the p-h asymmetry are the inter-layer gradient terms \(\Lambda \)  [4]}, [5]}, whose numerical value listed in Table REF  is about two times larger than the value given in Ref. [5]} and [4]}. As a consequence, compared with the BM model and other continuum theories, our effective theory for the microscopic model of Ref.[1]} leads to a much larger p-h asymmetry of the wavefunctions in the narrow bands.
